The Gerstenhaber Bracket As a Schouten Bracket for Polynomial Rings Extended by Finite Groups

THE GERSTENHABER BRACKET AS A SCHOUTEN BRACKET FOR POLYNOMIAL RINGS EXTENDED BY FINITE GROUPS

CRIS NEGRON AND SARAH WITHERSPOON

Abstract. We apply new techniques to compute Gerstenhaber brackets on the Hochschild cohomology of a skew group algebra formed from a polynomial ring and a ﬁnite group (in characteristic 0). We show that the Gerstenhaber brackets can always be expressed in terms of Schouten brackets on polyvector ﬁelds. We obtain as consequences some conditions under which brackets are always 0, and show that the Hochschild cohomology is a graded Gerstenhaber algebra under the codimension grading, strengthening known results.

1. Introduction We compute brackets on the Hochschild cohomology of a skew group algebra formed from a symmetric algebra (i.e. polynomial ring) and a finite group in characteristic 0. Our results strengthen those given by Halbout and Tang [8] and by Shepler and the second author [17]: The Hochschild cohomology decomposes as a direct sum indexed by conjugacy classes of the group. In [8, 17] the authors give formulas for Gerstenhaber brackets in terms of this decomposition, compute examples, and present vanishing results. Here we go further and show that brackets are always sums of projections of Schouten brackets onto these group components. As just one consequence, a bracket of two nonzero cohomology classes supported off the kernel of the group action is always 0 when their homological degrees are smallest possible. Our results complete the picture begun in [8, 17], facilitated here by new techniques from [13]. From a theoretical perspective, the category of modules over the skew group algebra under consideration here is equivalent to the category of equivariant quasi-coherent sheaves on the corresponding affine space. So the Hochschild cohomology of the skew group algebra, along with the Gerstenhaber bracket, is reflective of the deformation theory of this category. This relationship can be realized explicitly through the work of Lowen and Van den Bergh [11]. The Hochschild cohomology of the skew group algebra is also strongly related to Chen and Ruan’s orbifold cohomology (see e.g. [2, 15]). We briefly summarize our main results. If V is a finite dimensional vector space with an action of a finite group G, there is an induced action of G by automorphisms on the symmetric algebra S(V ), and one may form the skew group algebra (also known as a smash product or semidirect product) S(V )#G. Its Hochschild cohomology H :=

Date: May 25, 2017. The ﬁrst author was supported by NSF Postdoctoral fellowship DMS-1503147. The second author was supported by NSF grant DMS-1401016. 1 2 CRIS NEGRON AND SARAH WITHERSPOON

• HH (S(V )#G) is isomorphic to the G-invariant subspace of a direct sum ⊕g∈GHg, and the G-action permutes the components via the conjugation action of G on itself. See for example [4,7, 14]; we give some details as needed in Section 4.1. V• ∗ Each space Hg may be viewed in a canonical way as a subspace of S(V ) ⊗ V , V• ∗ and we construct canonical projections pg : S(V ) ⊗ V → Hg. Since the space S(V ) ⊗ V• V ∗ can be identified with the algebra of polyvector fields on affine space, it admits a canonical graded Lie bracket, namely the Schouten bracket (also known as the Schouten-Nijenhuis bracket), which we denote here by { , }. By way of the V• ∗ inclusions Hg ⊂ S(V ) ⊗ V we may apply the Schouten bracket to elements of Hg, • G and hence to elements in the Hochschild cohomology HH (S(V )#G) = (⊕g∈GHg) . Our main theorem is: P P • Theorem 5.2.3. Let X = g∈G Xg and Y = h∈G Yh be classes in HH (S(V )#G) where Xg ∈ Hg, Yh ∈ Hh. Their Gerstenhaber bracket is X [X,Y ] = pgh{Xg,Yh}. g,h∈G

This result was obtained by Halbout and Tang [8, Theorem 4.4, Corollary 4.11] in some special cases. V• ∗ In the body of the text, we assign to each summand Hg its own copy of S(V )⊗ V and label this copy with a g. So we will write instead S(V ) ⊗ V• V ∗g, and write V• ∗ elements in S(V ) ⊗ V g as Xgg instead of just Xg. P We note that all of the ingredients in the expressions pgh{Xg,Yh} are canonically defined. Thus we have closed-form expressions for Gerstenhaber brackets on classes in arbitrary degree. There are very few algebras for which we have such an understanding of the graded Lie structure on Hochschild cohomology, aside from smooth commutative algebras (over a given base field). For a smooth commutative algebra R there is the well known HKR isomorphism [9] between the Hochschild cohomology of R and polyvector fields on Spec(R), along with the Schouten bracket. The particular form of Theorem 5.2.3 is referential to this classic result. Having such a complete understanding of the Lie structure is useful, for example, in the production of L∞-morphisms and formality results (see [6, 10]). The proof of Theorem 5.2.3 uses the approach to Gerstenhaber brackets given in [13], in which we introduced new techniques that are particularly well-suited to computations. We summarize the necessary material from [13] in Section2, explaining how to apply it to skew group algebras. In Sections3 and4, we develop further the theory needed to apply the techniques to the skew group algebra S(V )#G in particular. We obtain a number of consequences of Theorem 5.2.3 in Sections5 and6. In Corollary 5.3.4 we recover [17, Corollary 7.4], stating that in case X,Y are supported entirely on group elements acting trivially on V , the Gerstenhaber bracket is simply the sum of the componentwise Schouten brackets. In Corollary 5.3.5 we recover [17, Proposition 8.4], giving some conditions on invariant subspaces under which the bracket [X,Y ] is known to vanish. Corollary 5.3.8 is another vanishing result that generalizes [17, Theorem 9.2] from degree 2 to arbitrary degree, stating that in case LIE BRACKET ON HOCHSCHILD COHOMOLOGY 3

X,Y are supported entirely off the kernel of the group action and their homological degrees are smallest possible, their Gerstenhaber bracket is 0. In Corollary 5.3.7 we also show that the Hochschild cohomology HH•(S(V )#G) is a graded Gerstenhaber algebra with respect to a certain natural grading coming from the geometry of the fixed spaces V g, which we refer to as the codimension grading. Section6 consists of examples and a general explanation of (non)vanishing of the Gerstenhaber bracket for S(V )#G, rephrasing some of the results of [17]. Let k be a field and ⊗ = ⊗k. For our main results, we assume the characteristic of k is 0, but this is not needed for the general techniques presented in Section2. We adopt the convention that a group G will act on the left of an algebra, and on the right of functions from that algebra. Similarly, we will let G act on the left of the finite dimensional vector space V ⊂ S(V ) and on the right of its dual space V ∗.

2. An alternate approach to the Lie bracket Let A be an algebra over the ﬁeld k. Let B → A denote the bar resolution of A as an A-bimodule: µ ··· −→δ3 A⊗4 −→δ2 A⊗3 −→δ1 A ⊗ A −→ A → 0, where µ denotes multiplication and n X i δn(a0 ⊗ · · · ⊗ an+1) = (−1) a0 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an+1 (2.0.1) i=0 for all a0, . . . , an+1 ∈ A. The Gerstenhaber bracket of homogeneous functions f, g ∈ HomAe (B,A) is deﬁned to be [f, g] = f ◦ g − (−1)(|f|−1)(|g|−1)g ◦ f, where the circle product f ◦ g is given by

(f ◦ g)(a1 ⊗ · · · ⊗ a|f|+|g|−1) |f| X (|g|−1)(j−1) = (−1) f(a1 ⊗ · · · ⊗ aj−1 ⊗ g(aj ⊗ · · · ⊗ aj+|g|−1) ⊗ · · · ⊗ a|f|+|g|−1), j=1 for all a1, . . . , a|f|+|g|−1 ∈ A, and similarly g◦f. This induces the bracket on Hochschild ⊗• cohomology. (Here we have identified HomAe (B,A) with Homk(A ,A).) Typically when one computes brackets, one uses explicit chain maps between this bar resolution B and a more convenient one for computational purposes, navigating back and forth. Such chain maps are usually awkward, and this way can be inefficient and technically difficult. In this section, we first recall from [13] an alternate approach, for some types of algebras, introduced to avoid this trouble. Then we explain how to apply it to skew group algebras in particular.

2.1. A collection of brackets. Given a bimodule resolution K → A satisfying some conditions as detailed below, one can produce a number of coarse brackets [ , ]φ on the complex HomAe (K,A), each depending on a map φ. These brackets are coarse in the sense that they will not, in general, produce dg Lie algebra structures on the complex 4 CRIS NEGRON AND SARAH WITHERSPOON

HomAe (K,A). They will, however, be good enough to compute the Gerstenhaber • • bracket on the cohomology H (HomAe (K,A)) = HH (A). We have precisely: Theorem 2.1.1 ([13, Theorem 3.2.5]). Suppose µ : K → A is a projective A-bimodule resolution of A satisfying the Hypotheses 2.1.2(a)–(c) below. Let FK : K ⊗A K → K be the chain map FK = µ ⊗ idK − idK ⊗µ. Then for any degree −1 bimodule map

φ : K ⊗A K → K satisfying dK φ + φdK⊗AK = FK , there is a bilinear operation [ , ]φ on HomAe (K,A). Each operation [ , ]φ satisﬁes the following properties. (1)[ f, g]φ is a cocycle whenever f and g are cocycles in HomAe (K,A). (2)[ f, g]φ is a coboundary whenever, additionally, f or g is a coboundary. • • (3) The induced operation on cohomology H (HomAe (K,A)) = HH (A) is precisely the Gerstenhaber bracket: [f, g]φ = [f, g] on cohomology.

The bilinear map [ , ]φ is deﬁned by equations (2.1.3) and (2.1.4) below. By [13, Lemma 3.2.1], such maps φ satisfying the conditions in the theorem always exist. We call such a map φ a contracting homotopy for FK . There is a diagonal map ∆B : B → B ⊗A B given by n X ∆B(a0 ⊗ ... ⊗ an+1) = (a0 ⊗ ... ⊗ ai ⊗ 1) ⊗A (1 ⊗ ai+1 ⊗ ... ⊗ an+1) i=0 for all a0, . . . , an+1 ∈ A. The hypotheses on the projective A-bimodule resolution K → A, to which the theorem above refers, are as follows. Hypotheses 2.1.2. (a) K admits a chain embedding ι : K → B that ﬁts into a commuting diagram ι K / B

% A y (b) The embedding ι admits a retract π. That is, there is an chain map π : B → K such that πι = idK . (c) The diagonal map ∆B : B → B ⊗A B preserves K, and hence restricts to a diagonal map ∆K : K → K ⊗A K. Equivalently, ∆Bι = (ι ⊗A ι)∆K . Conditions (a) and (c) together can alternatively be stated as the condition that K is a dg coalgebra in the monoidal category A-bimod that admits an embedding into the bar resolution. We denote the coproducts of elements in K using Sweedler’s notation, e.g.

∆(w) = w1 ⊗ w2, (∆ ⊗A idK )∆(w) = w1 ⊗ w2 ⊗ w3, for w ∈ K, with the implicit sum ∆(w) = P w ⊗ w suppressed. Koszul i1,i2 i1 i2 resolutions of Koszul algebras, as well as related algebras such as universal enveloping algebras, Weyl algebras, and Clifford algebras, will fit into this framework [1, 12]. Given a resolution K → A satisfying Hypotheses 2.1.2(a)–(c) and a contracting homotopy φ for FK , we construct the φ-bracket as follows: We first define the φ-circle product for functions f, g in HomAe (K,A) by

|w1||g| (f ◦φ g)(w) = (−1) f φ(w1 ⊗ g(w2) ⊗ w3) (2.1.3) LIE BRACKET ON HOCHSCHILD COHOMOLOGY 5 for homogeneous w in K, and define the φ-bracket as the graded commutator (|f|−1)(|g|−1) [f, g]φ = f ◦φ g − (−1) g ◦φ f. (2.1.4) We will be interested in producing such brackets particularly for a skew group algebra formed from a symmetric algebra (i.e. polynomial ring) under a finite group action. We will start with the Koszul resolution K of the symmetric algebra itself and then construct a natural extension Ke to resolve the skew group algebra. We will define a contracting homotopy φ for FK that extends to Ke, and use it to compute Gerstenhaber brackets via Theorem 2.1.1. We describe these constructions next in the context of more general skew group algebras.

2.2. Skew group algebras. Let G be a finite group whose order is not divisible by the characteristic of the field k. Assume that G acts by automorphisms on the algebra A. Let B = B(A) be the bar resolution of A, and let K = K(A) be a projective A- bimodule resolution of A satisfying Hypotheses 2.1.2(a)–(c). Assume that G acts on K and on B, and this action commutes with the differentials and with the maps ι, π, ∆K , ∆B. These assumptions all hold in the case that A is a Koszul algebra on which G acts by graded automorphisms and K is a Koszul resolution; in particular, 1 P −1 π may be replaced by |G| g∈G gπg if it is not a priori G-linear. Let A#G denote the skew group algebra, that is A ⊗ kG as a vector space, with multiplication defined by (a ⊗ g)(b ⊗ h) = a(gb) ⊗ gh for all a, b ∈ A and g, h ∈ G, where left superscript denotes the G-action. We will sometimes write a#g or simply ag in place of the element a ⊗ g of A#G when there can be no confusion. Let B(A#G) denote the bar resolution of A#G as a k-algebra, and let Be(A#G) denote its bar resolution over kG: µ δ3 ⊗kG4 δ2 ⊗kG3 δ1 ··· −→ (A#G) −→ (A#G) −→ (A#G) ⊗kG (A#G) −→ A#G → 0, with differentials defined as in (2.0.1). Since kG is semisimple, this is a projective resolution of A#G as an (A#G)e-module. There is a vector space isomorphism

(A#G)⊗kGi =∼ A⊗i ⊗ kG

⊗(j+2) for each i, and from now on we will identify Bej(A#G) with A ⊗ kG for each j, and the diﬀerentials of Be with those of B tensored with the identity map on kG. Similarly we wish to extend K to a projective resolution of A#G as an (A#G)e- module. Let Ke = Ke(A#G) denote the following complex:

d3⊗idkG d2⊗idkG d1⊗idkG µ⊗idkG ··· −−−−−→ K2 ⊗ kG −−−−−→ K1 ⊗ kG −−−−−→ K0 ⊗ kG −−−−→ A#G → 0. We give the terms of this complex the structure of A#G-bimodules as follows: (a#g)(x ⊗ h) = a(gx) ⊗ gh, (x ⊗ h)(a#g) = x(ha) ⊗ hg, for all a ∈ A, g, h ∈ G, and x ∈ Ki. Then Ke is a projective resolution of A#G by (A#G)e-modules. 6 CRIS NEGRON AND SARAH WITHERSPOON

Next we will show that Ke → A#G satisﬁes Hypotheses 2.1.2(a)–(c) for the algebra A#G. Let ˜ι : Ke → B(A#G) be the composition

ι⊗id i Ke −−−−→kG Be(A#G) −→ B(A#G) where i(a0 ⊗· · ·⊗aj+1 ⊗g) = (a0#1)⊗· · ·⊗(aj#1)⊗(aj+1#g) for all a0, . . . , aj+1 ∈ A and g ∈ G. Letπ ˜ : B(A#G) → Ke be the composition

p π⊗id B(A#G) −→ Be(A#G) −−−−→kG Ke where

p((a0#g0) ⊗ (a1#g1) ⊗ (a2#g2) ⊗ · · · ⊗ (aj#gj)) (2.2.1)

g0 g0g1 g0g1···gj−1 = a0 ⊗ ( a1) ⊗ ( a2) ⊗ · · · ⊗ ( aj) ⊗ g0g1 ··· gj for all a0, . . . , aj ∈ A and g0, . . . , gj ∈ G. One can check that i and p are indeed chain maps. Then π˜˜ι = (π ⊗ id )pi(ι ⊗ id ) = id kG kG Ke since pi = id by the definitions of these maps, and πι = id by Hypothe- Be(A#G) K sis 2.1.2(b) applied to K. Therefore Hypotheses 2.1.2(a) and (b) hold for Ke. Now let ∆ : K → K ⊗ K be defined by ∆ = ∆ ⊗ id , after identifying Ke e e A#G e Ke K kG Kei ⊗A#G Kej = (Ki ⊗ kG) ⊗A#G (Kj ⊗ kG) with (Ki ⊗A Kj) ⊗ kG. One may check that ∆ satisfies Hypothesis 2.1.2(c). Ke Let φK : K ⊗A K → K be a map satisfying dK φK + φK dK⊗AK = FK as in Theorem 2.1.1. Assume that φK is G-linear. Let

φ = φ ⊗ id , (2.2.2) Ke K kG ∼ a map from Ke ⊗A#G Ke to Ke, under the identiﬁcation Ke ⊗A#G Ke = (K ⊗A K) ⊗ kG. Since φ is G-linear, this map φ is an A#G-bimodule map. Further, K Ke

d φ + φ d = (dK φK + φK dK⊗ K ) ⊗ idkG = FK ⊗ idkG = F . Ke Ke Ke Ke⊗AKe A Ke As a consequence, by Theorem 2.1.1, φ may be used to deﬁne the Gerstenhaber Ke bracket on the Hochschild cohomology of A#G via (2.1.3) and (2.1.4).

3. Symmetric algebras Let V be a finite dimensional vector space over the field k of characteristic 0, and let A = S(V ), the symmetric algebra on V . In this section, we construct a map φ that will allow us to compute Gerstenhaber brackets on the Hochschild cohomology of the skew group algebra A#G arising from a representation of the finite group G on V , via Theorem 2.1.1 and equation (2.2.2). LIE BRACKET ON HOCHSCHILD COHOMOLOGY 7

3.1. The Koszul resolution. We will use a standard description of the Koszul resolution K of A = S(V ) as an Ae-bimodule, given as a subcomplex of the bar resolution B = B(A): For all v1, . . . , vi ∈ V , let X o(v1, . . . , vi) = sgn(σ)vσ(1) ⊗ · · · ⊗ vσ(i) σ∈Si in V ⊗i ⊂ A⊗i. Sometimes we write instead

o(vI ) = o(v1, . . . , vi), e where I = {1, . . . , i}. Take o(∅) = 1. Let Ki = Ki(A) be the free A -submodule ⊗(i+2) ⊗(i+2) of A with free basis all 1 ⊗ o(v1, . . . , vi) ⊗ 1 in A . We may in this way Vi V• identify Ki with A ⊗ V ⊗ A and K with A ⊗ V ⊗ A. The diﬀerentials on the bar resolution, restricted to K, may be rewritten in terms of the chosen free basis of K as d(1 ⊗ o(v1, . . . , vi) ⊗ 1) i X j−1 = (−1) (vj ⊗ o(v1,..., vˆj, . . . , vi) ⊗ 1 − 1 ⊗ o(v1,..., vˆj, . . . , vi) ⊗ vj), j=1 wherev ˆj indicates that vj has been deleted from the list of vectors. Note that the action of G preserves the vector subspace of Ki spanned by all 1 ⊗ o(v1, . . . , vi) ⊗ 1. There is a retract π : B → K that can be chosen to be G-linear. We will not need an explicit formula for π here. The diagonal map ∆K : K → K ⊗A K is given by X ∆K (1 ⊗ o(vI ) ⊗ 1) = ±(1 ⊗ o(vI1 ) ⊗ 1) ⊗ (1 ⊗ o(vI2 ) ⊗ 1) (3.1.1) I1,I2 where the sum is indexed by all ordered disjoint subsets I1,I2 ⊂ I with I1 ∪ I2 = I, and ± is the sign of σ, the unique permutation for which {iσ(1), . . . iσ(|I|)} = I1 ∪ I2 as an ordered set. Note that ∆K is G-linear. Letting ι : K → B be the embedding of K as a subcomplex of B, Hypotheses 2.1.2(a)–(c) now hold. We may thus use Theorem 2.1.1 and (2.2.2) to compute brackets on HH•(S(V )#G) once we have a suitable map φ.

3.2. An invariant map φ. We will now define an A-bilinear map φ : K ⊗A K → K that will be G-linear and independent of choice of ordered basis of V . Compare with [13, Definition 4.1.3], where a simpler map φ was defined which however depends on such a choice. At first glance, the map φ below looks rather complicated, but in practice we find it easier to use when extended to a skew group algebra than explicit chain maps between bar and Koszul resolutions.

Deﬁnition 3.2.1. Assume the characteristic of k is 0. Let φ : K ⊗A K → K be the A-bilinear map given on ordered monomials as follows:

1 X φ (1 ⊗ v ··· v ⊗ 1) = v ··· v ⊗ o(v ) ⊗ v ··· v 0 1 t t! σ(1) σ(r−1) σ(r) σ(r+1) σ(t) 1≤r≤t σ∈St 8 CRIS NEGRON AND SARAH WITHERSPOON for all v1, . . . , vt ∈ V . On K0 ⊗A Kz and on Ks ⊗A K0 (s, z > 0), let

φz(1 ⊗ v1 ··· vt ⊗ o(w1, . . . , wz) ⊗ 1) z−1 (−1)z X Y = (r + i)v ··· v ⊗ o(w , . . . , w , v ) ⊗ v ··· v , (t + z)! σ(1) σ(r−1) 1 z σ(r) σ(r+1) σ(t) 1≤r≤t i=0 σ∈St φs(1 ⊗ o(u1, . . . , us) ⊗ v1 ··· vt ⊗ 1 s 1 X Y = (t − r + i)v ··· v ⊗ o(v , u , . . . , u ) ⊗ v ··· v , (t + s)! σ(1) σ(r−2) σ(r) 1 s σ(r+1) σ(t) 1≤r≤t i=1 σ∈St for all u1, . . . , us, v1, . . . , vt, w1, . . . , wz ∈ V . When s > 0 and z > 0, let

φs+z(1 ⊗ o(u1, . . . , us) ⊗ v1 ··· vt ⊗ o(w1, . . . , wz) ⊗ 1) X s,t,z = cr vσ(1) ··· vσ(r−1) ⊗ o(w1, . . . , wz, vσ(r), u1, . . . , us) ⊗ vσ(r+1) ··· vσ(t), 1≤r≤t σ∈St z−1 s (−1)sz+z Y Y where cs,t,z = (r + i) (t − r + j). r (s + t + z)! i=0 j=1 To see that φ is well-defined, one can first construct the corresponding map from the tensor powers (V ⊗s) ⊗ (V ⊗t) ⊗ (V ⊗z) (using the universal property of the tensor product, for example) then note that the given map is Ss × St × Sz-invariant and hence induces a well-defined map φ on the coinvariants (Vs V )⊗St(V )⊗(Vz V ). One can also see directly that φ is G-invariant. In fact, it is invariant under the action of the entire group GL(V ). We next state that φ is a contracting homotopy for the map FK defined in the statement of Theorem 2.1.1.

Lemma 3.2.2. Let φ : K ⊗A K → K be the A-bilinear map of Deﬁnition 3.2.1. Then dK φ + φdK⊗AK = FK . Proof. In degree 0, we check:

(dφ + φd)(1 ⊗ v1 ··· vt ⊗ 1)    1 X  = d  vσ(1) ··· vσ(r−1) ⊗ o(vσ(r)) ⊗ vσ(r+1) ··· vσ(t) t!  1≤r≤t  σ∈St 1 X = (v ··· v ⊗ v ··· v − v ··· v ⊗ v ··· v ) t! σ(1) σ(r) σ(r+1) σ(t) σ(1) σ(r−1) σ(r) σ(t) 1≤r≤t σ∈St 1 X = (v ··· v ⊗ 1 − 1 ⊗ v ··· v ) t! σ(1) σ(t) σ(1) σ(t) σ∈St = v1 ··· vt ⊗ 1 − 1 ⊗ v1 ··· vt = FK (1 ⊗ v1 ··· vt ⊗ 1). Other veriﬁcations are tedious, but similar. LIE BRACKET ON HOCHSCHILD COHOMOLOGY 9

4. φ-circle product formula and projections onto group components We first recall some basic facts about the Hochschild cohomology of the skew group algebra S(V )#G. The graded vector space structure of the cohomology is well-known, see for example [4,7, 14]. We give some details here as will be needed for our bracket computations. We then derive a formula for φ-circle products (defined in (2.1.3)) on this Hochschild cohomology, and define projection operators needed for our main results. We assume from now on that the characteristic of k is 0. Then HH•(S(V )#G) =∼ HH•(S(V ),S(V )#G)G, (4.0.1) where the superscript G denotes invariants of the action of G on Hochschild cohomology induced by its action on complexes (via the standard group action on tensor products and functions). This follows for example from [5]. We will focus our discussions and computations on HH•(S(V ),S(V )#G), and results for Hochschild cohomology HH•(S(V )#G) will follow by restricting to its G-invariant subalgebra.

• 4.1. Structure of the cohomology HH (S(V ),S(V )#G). Let {x1, . . . , xn} be a ∗ ∗ ∗ basis for V and {x1, . . . , xn} the dual basis for its dual space V . The Hochschild cohomology HH•(S(V ),S(V )#G) is computed as the homology of the complex V• ∼ V• HomS(V )e (S(V ) ⊗ V ⊗ S(V ),S(V )#G) = Homk( V,S(V )#G) M =∼ S(V ) ⊗ V•V ∗g. g∈G L V• ∗ The diﬀerential on the graded space g∈G S(V ) ⊗ V g induced by the above sequence of isomorphisms is left multiplication by the diagonal matrix

X g E = diag{Eg}g∈G,Eg = (xi − xi)∂i, i ∗ P P where ∂i = 1 ⊗ xi . So E · ( g Ygg) = g(EgYg)g. This complex breaks up into a V• ∗ sum of subcomplexes (S(V ) ⊗ V g, Eg), and we have M HH•(S(V ),S(V )#G) = H•(S(V ) ⊗ V•V ∗g). g∈G

We note that each Eg is independent of the choice of basis, since it is simply the image V0 V• of 1g ∈ Homk( V,S(V )g) under the diﬀerential on Homk( V,S(V )g). −1 g The right G-action on HomS(V )e (K,S(V )#G) is given by (f · g)(x) = g f( x)g, for f ∈ HomS(V )e (K,S(V )#G) and g ∈ G, giving it the structure of a G-complex. This translates to the action h−1 h h −1 (a ⊗ f1 . . . fl)g) · h = ( a) ⊗ f1 . . . fl h gh (4.1.1) L V• ∗ ∗ on g∈G S(V ) ⊗ V g, where a ∈ S(V ) and fi ∈ V . It will be helpful to have the following general lemma. Lemma 4.1.2. Given any G-representation M, and element g ∈ G, there is a canonical complement to the g-invariant subspace M g, given by (M g)⊥ = (1 − g) · M. This 10 CRIS NEGRON AND SARAH WITHERSPOON gives a splitting M = M g ⊕(1−g)M of M as a hgi-representation. This decomposition satisﬁes −1 M g = M g and (1 − g)M = (1 − g−1)M and is compatible with the G-action in the sense that, for any h ∈ G,

−1 h · M g = M hgh and h · (1 − g)M = (1 − hgh−1)M. Proof. The operation (1 − g) · − : M → M has kernel precisely M g. Furthermore M g ∩ (1 − g) · M = 0, since for any invariant element m − gm in (1 − g) · M we will have Z Z Z Z Z (m − gm) = (m − gm) = m − gm = m − m = 0, g g g g g R 1 P|g|−1 i g where g = |g| i=0 g . We conclude that M = M ⊕ (1 − g)M when M is finite dimensional, by a dimension count. When M is infinite dimensional, the result can be deduced from the fact that M is the union of its finite dimensional submodules. The equality M g = M g−1 is clear, and the equality (1 − g)M = (1 − g−1)M follows from the fact that M = −gM. As for the compatibility claim, the identity h · M g = M hgh−1 is obvious, while the equality h(1 − g)M = (1 − hgh−1)M follows from the computation h(1 − g)M = h(1 − g)h−1M = (hh−1 − hgh−1)M = (1 − hgh−1)M.

⊥ Let us take detg to be the one dimensional hgi-representation: ⊥ VcodimV g ∗ detg = ((1 − g)V ) g. We then have the embedding g V•−codimV g g ∗ ⊥ V• ∗ S(V ) ⊗ (V ) detg −→ S(V ) ⊗ V g (4.1.3) induced by the embedding of V g as a subspace of V and a corresponding dual subspace embedding. Lemma 4.1.4. For each g ∈ G, g V•−codimV g g ∗ ⊥ Eg · (S(V ) ⊗ (V ) detg ) = 0 and g V•−codimV g g ∗ ⊥ (Im(Eg · −)) ∩ S(V ) ⊗ (V ) detg = 0. g V•−codimV g g ∗ ⊥ That is to say, the subspaces S(V )⊗ (V ) detg consist entirely of cocycles and contain no nonzero coboundaries.

Proof. If we choose a basis {x1, . . . , xn} for V such that the ﬁrst l elements are a basis for V g, and the remaining are a basis for (1 − g)V , then we have n X g X g Eg = (xi − xi)∂i = (xi − xi)∂i, i=1 i>l LIE BRACKET ON HOCHSCHILD COHOMOLOGY 11

g ⊥ since xi = xi for all i ≤ l. So Eg detg = 0 and, if we let Ig denote the ideal in S(V ) generated by (1 − g)V , we have V• ∗ V• ∗ Eg (S(V ) ⊗ V g) ⊂ Ig ⊗ V g. The second statement here implies that the proposed intersection is trivial. By the above information, there is an induced map g V•−codimV g g ∗ ⊥ • V• ∗ S(V ) ⊗ (V ) detg −→ H S(V ) ⊗ V g which is injective. The following is a rephrasing of Farinati’s calculation [4]. Proposition 4.1.5. The induced maps g V•−codimV g g ∗ ⊥ • V• ∗ S(V ) ⊗ (V ) detg −→ H S(V ) ⊗ V g are isomorphisms for each g ∈ G, and so there is an isomorphism M g V•−codimV g g ∗ ⊥ =∼ • M V• ∗ S(V ) ⊗ (V ) detg −→ H (S(V ) ⊗ V g) . (4.1.6) g∈G g∈G Recalling that the codomain of (4.1.6) is the cohomology HH•(S(V ),S(V )#G), the second portion of this proposition gives an identiﬁcation M g V•−codimV g g ∗ ⊥ • S(V ) ⊗ (V ) detg = HH (S(V ),S(V )#G). (4.1.7) g∈G In addition to providing this description of the cohomology, the embedding (4.1.3) is compatible with the G-action in the following sense. Proposition 4.1.8. (1) For any g, h ∈ G there is an equality

−1 g V•−codimV g g ∗ ⊥ h−1gh V•−codimV h gh h−1gh ∗ ⊥ S(V ) ⊗ (V ) detg · h = S(V ) ⊗ (V ) deth−1gh L V• ∗ in g S(V ) ⊗ V g. L g V•−codimV g g ∗ ⊥ (2) The sum g S(V ) ⊗ (V ) detg is a G-subcomplex of the sum L V• ∗ g S(V ) ⊗ V g. (3) The isomorphism M g V•−codimV g g ∗ ⊥ =∼ • M V• ∗ S(V ) ⊗ (V ) detg −→ H (S(V ) ⊗ V g) g∈G g∈G is one of graded G-modules. Proof. From Lemma 4.1.2, and the descriptions of (V g)∗ and ((1 − g)V )∗ as those functions vanishing on (1 − g)V and V g respectively, we have

−1 (V g)∗ · h = {functions vanishing on h−1(1 − g)V } = (V h gh)∗ ((1 − g)V )∗ · h = {functions vanishing on h−1V g} = ((1 − h−1gh)V )∗ and h−1 · S(V g) = S(V h−1gh). This, along with the description (4.1.1) gives the equality in (1). Statement (2) follows from (1), and (3) follows from (2) and Propo- sition 4.1.5. 12 CRIS NEGRON AND SARAH WITHERSPOON

4.2. The φ-circle product formula. We will compute ﬁrst with the complex M V• ∗ HomS(V )e (K,S(V )#G) = S(V ) ⊗ V g, g∈G and then restrict to G-invariant elements to make conclusions about the cohomology HH•(S(V )#G). Letting φ be the map of Deﬁnition 3.2.1 and φ = φ ⊗ id as K Ke K kG in equation (2.2.2), φ := φ gives rise to a perfectly good bilinear operation Ke

|w1||Y | h Xg ◦φ Y h = w 7→ (−1) Xg(φ(w1 ⊗ Y (w2) ⊗ w3))h , which need not be a chain map. Here X,Y ∈ S(V ) ⊗ V•V ∗, and w ∈ K. We can also deﬁne the φ-bracket in the most naive manner as (|X|−1)(|Y |−1) [Xg, Y h]φ = Xg ◦φ Y h − (−1) Y h ◦φ Xg. This operation, again, need not be well behaved at all on non-invariant functions, but it will be a bilinear map. In the lemma below, we give a formula for the φ-circle product of special types of elements. Our formula may be compared with [8, Lemma 4.1] and [17, Theorem 7.2]. Due to the structure of the Hochschild cohomology of S(V )#G stated in Proposi- tion 4.1.5, this formula will in fact suﬃce to compute all brackets. To do this, we will only need to consider G-invariant elements and compute relevant φ-circle products on summands representing elements in HH•(S(V ),S(V )g) × HH•(S(V ),S(V )h) for all pairs g, h in G. This we will do in Section5.

VcodimV h ∗ Lemma 4.2.1. Let g, h ∈ G, p ≥ 0, and ωh any generator for ((1 − h)V ) . ∗ Vp ∗ Consider any Xi ∈ S(V ) ⊗ V , Y¯ ∈ V ωh, and Y = v1 . . . vtY¯ for vi ∈ V . Then

(X1 ...Xdg ◦φ Y h) = X l,d,t g g ¯ ζr,|Y |vσ(1) . . . vσ(r−1)Xl(vσ(r)) vσ(r+1) ... vσ(t)(X1 ...Xl−1)Y (Xl+1 ...Xd)gh l,r,σ l,d,t where the sum is over 1 ≤ l ≤ d, 1 ≤ r ≤ t, σ ∈ St, and the coeﬃcients ζr,m are the nonzero rational numbers (r + l − 2)!(t − r + d − l)! ζl,d,t = (−1)(m−1)(l−1) . r,m (r − 1)!(t − r)!(d + t + l − 1)!

h ⊥ h ∗ ∗ Proof. Choose bases y1, . . . , ys of (V ) and ys+1, . . . , yn of V . Let y1, . . . , yn be the ∗ ∗ dual basis of V . Recall the notation ∂j = 1 ⊗ yj . We may write each of the Xi as a sum of elements of the form f∂l, with f ∈ S(V ), and write Y¯ as a sum of elements of the form ∂1 . . . ∂s∂J , with J an ordered subset of {s + 1, . . . , n}. By expanding both sides of the proposed equality accordingly, one sees that it suﬃces to prove the result in the case in which the X are of the form f ∂∗ and Y¯ = ∂ . . . ∂ ∂ . We are free to i i ji 1 s J reorder indices if necessary so that Y¯ = ∂1 . . . ∂m with m ≥ s. Let us check the value of the φ-circle product when applied to monomials of the form o(yjl+1 , . . . , yjd , y1, . . . , ym, yj1 , . . . , yjl−1 ), for arbitrary l. If any of the yji are in (V h)⊥, or if any of the indices are repeated, then the given monomial will be 0. So LIE BRACKET ON HOCHSCHILD COHOMOLOGY 13

h we may assume that each of the yji , for i 6= l, are in V and that none of the indices are repeated. Applying the definition (2.1.3) of the φ-circle product and the diagonal map (3.1.1), we find ((X1 ··· Xdg) ◦φ (Y h)) o(yjl+1 , . . . , yjd , y1, . . . , ym, yj1 , . . . , yjl−1 ) (d−l)m h = (−1) (X1 ··· Xdg) φ(o(yjl+1 , . . . , yjd ) ⊗ v1 ··· vt ⊗ o(yj1 , . . . , yjl−1 ))h . h h Since all the yji are in V , we may replace o(yj1 , . . . , yjl−1 ) by o(yj1 , . . . , yjl−1 ) in the above expression. Therefore the value of (X1 ··· Xdg) ◦φ (Y h) on the given monomial is as follows, using Definition 3.2.1 of φ: (d−l)m (−1) (X1 ··· Xdg) φ(o(yjl+1 , . . . , yjd ) ⊗ v1 ··· vt ⊗ o(yj1 , . . . , yjl−1 )h (d−l)m X d−l,t,l−1 = (−1) (X1 ··· Xdg) cr vσ(1) ··· vσ(r−1)⊗ 1≤r≤t σ∈St o(yj1 , . . . , yjl−1 , vσ(r), yjl+1 , . . . , yjd ) ⊗ vσ(r+1) ··· vσ(t)h (d−l)m X d−l,t,l−1 = (−1) cr vσ(1) ··· vσ(r−1)Xl(vσ(r))f1 ··· fl−1fl+1 ··· fdgvσ(r+1) ··· vσ(t)h. 1≤r≤t σ∈St The above argument can also be used to show that the circle product vanishes on all monomials o(yI ) which are not of the form ±o(yj1 , . . . , yjl−1 , y1, . . . , ym, yjl+1 , . . . , yjd ) for some l. To obtain the equality of the theorem, we must reorder the factors in our initial argument:

o(yjl+1 , . . . , yjd , y1, . . . , ym, yj1 , . . . , yjl−1 ) m(l−1)+(d−l)(l−1)+(d−l)m = (−1) o(yj1 , . . . , yjl−1 , y1, . . . , ym, yjl+1 , . . . , yjd ). Multiply by this coefficient and compare values with those of the function in the statement of the theorem to see that they are the same. 4.3. Projections onto group components. For each g ∈ G, we will construct a chain retraction V• ∗ g V•−codimV g g ∗ ⊥ pg : S(V ) ⊗ V g −→ S(V ) ⊗ (V ) detg g V•−codimV g g ∗ ⊥ onto the subspace S(V )⊗ (V ) detg . (The differential on the codomain is taken to be 0.) Simply by virtue of being a retract of an injective quasi-isomorphism, each pg will also be a quasi-isomorphism. In the sections that follow we will often think of the pg as quasi-isomorphisms from S(V ) ⊗ V•V ∗g to itself, simply by composing with the embedding. We outline the construction of pg below. g Construction of pg. From the canonical decomposition V = V ⊕(1−g)V we get an g 1 g identification S(V ) = S(V ⊕ (1 − g)V ) and canonical projection pg : S(V ) → S(V ). We also get a canonical decomposition of the dual space and its higher wedge powers, M V ∗ = (V g)∗ ⊕ ((1 − g)V )∗ and ViV ∗ = (Vi1 (V g)∗) ∧ (Vi2 ((1 − g)V )∗),

i1+i2=i 14 CRIS NEGRON AND SARAH WITHERSPOON whence we have a second canonical projection 2 V• ∗ V•−codimV g g ∗ VcodimV g ∗ V•−codimV g g ∗ ⊥ pg : V → (V ) ∧ ( ((1 − g)V ) ) = (V ) detg . 1 2 We now deﬁne pg as the tensor product pg ⊗ pg, V• ∗ g V•−codimV g g ∗ ⊥ pg : S(V ) ⊗ V g −→ S(V ) ⊗ (V ) detg . (4.3.1)

It is apparent from the construction that each pg restricts to the identity on the g V•−codimV g g ∗ ⊥ subspace S(V ) ⊗ (V ) detg . Furthermore, since the ideal Ig generated 1 by (1 − g)V is precisely the kernel of pg, and left multiplication by Eg has image in V• ∗ Ig ⊗ V , we see that pg(Eg ·−) = 0. This is exactly the statement that pg is a chain map. Note that, by Proposition 4.1.5, the projections will be quasi-isomorphisms. Recall that, by Proposition 4.1.8, the subspace M g V•−codimV g g ∗ ⊥ M V• ∗ S(V ) ⊗ (V ) detg ⊂ S(V ) ⊗ V g g g is a G-subcomplex. These projections pg are compatible with the G-action in the sense of V• ∗ Lemma 4.3.2. (1) For any Xg ∈ S(V ) ⊗ V g the projections pg and ph−1gh satisfy the relation ph−1gh((Xg) · h = pg(Xg) · h. (2) The coproduct map M V• ∗ M g V•−codimV g g ∗ ⊥ p : S(V ) ⊗ V g −→ S(V ) ⊗ (V ) detg , g g

i.e. the map p = diag{pg : g ∈ G}, is a G-linear quasi-isomorphism. P P (3) If a sum of elements g Xgg is G-invariant then so is g pg(Xgg). Proof. As was the case in the proof of Proposition 4.1.8, (1) follows from the com- patibilities of the decompositions V = V g ⊕ (1 − g)V with the G-action given in Lemma 4.1.2. Statements (2) and (3) follow from (1) and the fact that each pg is a quasi-isomorphism.

In the following results, we take Ig ⊂ S(V ) to be the ideal generated by (1 − g)V , as was done above. Given an ordered subset I = {i1, . . . , ij} of {1, . . . , n}, let ∂I = V• ∗ ∗ ∂i1 ··· ∂ij in S(V ) ⊗ V where ∂i = 1 ⊗ xi as before. VcodimV g ∗ Lemma 4.3.3. Let ωg be an arbitrary generator for ((1 − g)V ) . The kernel of the projection pg deﬁned in (4.3.1) is the sum V• ∗ ker(pg) = Ig ⊗ V g + S(V ) ·{∂I g : ωg does not divide ∂I }. g If we choose bases {x1, . . . , xl} of V and {xl+1, . . . , xn} of (1 − g)V , and take ωg = ∂l+1 ··· ∂n, the second set can be written as

S(V ) ·{∂I g : {l + 1, . . . , n} is not a subset of I}. LIE BRACKET ON HOCHSCHILD COHOMOLOGY 15

1 2 Proof. Note that, for pg and pg as in the above construction of pg, we have 1 2 ker(pg) = Ig and ker(pg) = k{∂I g : ωg does not divide ∂I }

So the description of ker(pg) follows from the fact that for any product of vector space maps σ1⊗σ2 : W1⊗W2 → U1⊗U2, its kernel is the sum ker(σ1)⊗W2+W1⊗ker(σ2).

5. Brackets In this section we assume the characteristic of k is 0, and derive a general formula for brackets on Hochschild cohomology of S(V )#G, using the φ-circle product formula of Lemma 4.2.1 and the projection maps (4.3.1). The Schouten bracket for the underlying symmetric algebra features prominently. We use our formula to obtain several conclusions about brackets, in particular some vanishing criteria. We will use the notation and results of Section4. In particular, we will express elements of the Hochschild cohomology HH•(S(V )#G) as G-invariant elements of HH•(S(V ),S(V )#G) by (4.0.1), and we will use the identiﬁcation of cohomology, M g V•−codimV g g ∗ ⊥ • S(V ) ⊗ (V ) detg = HH (S(V ),S(V )#G) g∈G given by (4.1.7). Elements of cohomology HH•(S(V ),S(V )#G) that are nonzero only in the component indexed by a unique g in G in the above sum are said to be supported on g. Elements that are nonzero only in components indexed by elements in the conjugacy class of g are said to be supported on the conjugacy class of g. The V• ∗ g V• g ∗ ⊥ canonical projections pg : S(V )⊗ V g → S(V )⊗ (V ) detg g, deﬁned in (4.3.1), will appear in our expressions of brackets on Hochschild cohomology HH•(S(V )#G) below.

5.1. Preliminary information on the Schouten bracket and group actions. We let the bracket {X,Y } denote the standard Schouten bracket on S(V ) ⊗ V•V ∗ =∼ V• S(V )T , where T denotes the global vector ﬁelds on Spec(S(V )) (or rather, algebra derivations on S(V )). Gerstenhaber brackets in this case are precisely Schouten brackets, and we brieﬂy verify that our approach does indeed give this expected result: Lemma 5.1.1. For any X,Y ∈ S(V ) ⊗ V•V ∗,

[X,Y ]φ = {X,Y }. ˜ ˜ Proof. By construction, the restriction of φ to K ⊗S(V ) K ⊂ K ⊗S(V )#G K provides a V• ∗ contracting homotopy for FK . Therefore, for elements in S(V ) ⊗ V , we will have [X,Y ] = [X,Y ] . So it does make sense to consider the proposed equality φ φ|K⊗S(V )K V• ∗ ∼ V• [X,Y ]φ = {X,Y } in S(V ) ⊗ V = S(V )T . It is shown in [13, §4] that some choice of contracting homotopy ψ is such that [X,Y ]ψ is the Schouten bracket. By [13, Theorem 3.2.5], for any two choices of contracting homotopy, the diﬀerence in the associated brackets is a coboundary. Since V• ∗ the diﬀerential vanishes on HomAe (K,S(V )) = S(V ) ⊗ V , we see that [X,Y ]φ = [X,Y ]ψ = {X,Y }. 16 CRIS NEGRON AND SARAH WITHERSPOON

V• ∗ ∼ V• Under the identification S(V ) ⊗ V = S(V )T , the right action of an element g ∈ G on S(V )⊗V ∗ is identified with conjugation by the corresponding automorphism, X · g = Xg = g−1Xg. On higher degree elements the action is given by the standard g g g formula, (X1 ··· Xl) · g = (X1 ··· Xl) = X1 ··· Xl . Under the identification M V• ∗ ∼ M V• (S(V ) ⊗ V g) = ( S(V )T g), g∈G g∈G the action is given by (Xg) · h = Xhh−1gh, where we can view X either as an element in S(V ) ⊗ V ∗ or as a polyvector field. Lemma 5.1.2. For any X,Y ∈ S(V ) ⊗ V•V ∗, and g ∈ G, we have {X,Y }g = {Xg,Y g}. V• ∗ V• Proof. We identify S(V ) ⊗ V with the global polyvector fields S(V )T . Then the lemma follows from the fact that the G-action is simply given by conjugating by the corresponding automorphism, the fact that the Schouten bracket is given by composition of vector fields on T , and the Gerstenhaber identity {X,Y1Y2} = {X,Y1}Y2 ± Y1{X,Y2}. P P Lemma 5.1.3. For any G-invariant elements g Xgg and h∈G Yhh in the sum L V• ∗ P g∈G(S(V ) ⊗ V g), the element g,h∈G{Xg,Yh}gh is also G-invariant. Further- P P P more, for any such g Xgg and h Yhh, the element g,h∈G pgh{Xg,Yh}gh will be a G-invariant cocycle. P Proof. We know an element g Zgg will be invariant if and only if, for each g, σ ∈ G, σ Zg = Zσ−1gσ. So the Xg and Yh have this property, and it follows that the sum P {g,h∈G:gh=τ}{Xg,Yh} will have this property for each τ ∈ G since X σ X σ σ X X {Xg,Yh} = {Xg ,Yh } = {Xσ−1gσ,Yσ−1hσ} = {Xg0 ,Yh0 }. gh=τ gh=τ gh=τ {g0,h0:g0h0=στσ−1} The last statement now follows directly from Lemma 4.3.2(3). 5.2. φ-brackets as Schouten brackets. Before we begin, it will be useful to have the following two lemmas. Recall that Ig is the ideal generated by (1 − g)V in S(V ). (1−g) Lemma 5.2.1. For any vectors ui1 , . . . , uiν ∈ V and g ∈ G, the element (ui1 . . . uiν ) is in Ig. Proof. We proceed by induction on the number of vectors ν. When ν = 1 the result is immediate. For ν > 1 we have (1−g) g g (ui1 . . . uiν ) = ui1 . . . uiν − ui1 ... uiν g g g g (1−g) = (ui1 . . . uiν−1 − ui1 ... uiν−1 )uiν + ui1 ... uiν−1 uiν , which is now in Ig by induction. 0 L V• ∗ Lemma 5.2.2. Suppose c and c are invariant cocycles in g S(V ) ⊗ V g that differ by a (possibly non-invariant) coboundary. Then c and c0 differ by an invariant coboundary. LIE BRACKET ON HOCHSCHILD COHOMOLOGY 17

Proof. Note that c − c0 is also invariant. Let b be any element with d(b) = c − c0, where d is the differential d = E · −. Then we have Z Z c − c0 = d(b) · = d(b · ), G G R 1 P R where G is the standard integral |G| g∈G g. So b · G provides the desired invariant bounding element. We can now give a general formula for the Gerstenhaber bracket on HH•(S(V )#G) in terms of Schouten brackets. One may compare with [8, Theorem 4.4, Corollary 4.11] where the authors give similar formulas under some conditions on the group G and its action on V . P P Theorem 5.2.3. Let X = g Xgg and Y = h Yhh be G-invariant cocycles in g V•−codimV g g ∗ ⊥ P ⊕g∈GS(V )⊗ (V ) detg . The sum g,h∈G pgh{Xg,Yh}gh is a G-invariant cocycle and, considered as elements of the cohomology HH•(S(V )#G), X [X,Y ] = pgh{Xg,Yh}gh. g,h∈G Proof. By Lemma 5.2.2, it suffices to show that the equality holds up to arbitrary coboundaries. Note that X (|Y |−1)(|X|−1) X p[X,Y ]φ = pgh(Xgg ◦φ Yhh) − (−1) phg(Yhh ◦φ Xgg). g,h g,h By considering the group automorphism G × G → G × G, (g, h) 7→ (g, ghg−1), and the equality ghg−1g = gh, we see that we can reindex the second sum to obtain P P −1 p[X,Y ]φ = g,h pgh(Xgg ◦φ Yhh) ∓ g,h pgh(Yghg−1 ghg ◦φ Xgg) P −1 (5.2.4) = g,h pgh(Xgg ◦φ Yhh) ∓ pgh(Yghg−1 ghg ◦φ Xgg) . We claim that there is an equality −1 pgh(Xgg◦φYhh)∓pgh(Yghg−1 ghg ◦φXgg) = pgh[Xg,Yh]φgh = pgh{Xg,Yh}gh (5.2.5) for each pair g, h ∈ G, where the second equality follows already by Lemma 5.1.1. If we can establish (5.2.5) then we are done, by the final expression in (5.2.4) and the fact that the difference [X,Y ]φ − p[X,Y ]φ is a coboundary. Let us fix elements g, h ∈ G. Write Xg as a sum of elements of the form u1 ··· usX¯ g h with the uj ∈ V , and Yh as a sum of elements v1 ··· vtY¯ with the vi ∈ V . Here V• ∗ g gh X,¯ Y¯ ∈ V . By h-invariance there is an equality vi = vi for each i. For arbitrary V• ∗ (1−gh) Z in S(V ) ⊗ V and elements ai ∈ S(V ), the projection pgh( (a1 . . . aq)Zgh) vanishes by Lemma 4.3.3 and Lemma 5.2.1. So for any r < t and σ ∈ Sn there will be equalities p g(v . . . v )Zgh = p gh(v . . . v )Zgh gh σ(r+1) σ(t) gh σ(r+1) σ(t) (5.2.6) = pgh(vσ(r+1) . . . vσ(t)Zgh). 18 CRIS NEGRON AND SARAH WITHERSPOON

It now follows from the expression for the circle operation given in Lemma 4.2.1 that there is an equality

pgh(Xgg ◦φ Yhh) = pgh(Xg ◦φ Yh)gh. (5.2.7) This covers half of what we need. We would like to show now −1 pgh(Yghg−1 ghg ◦φ Xgg) = pgh(Yh ◦φ Xg). −1 Simply replacing g and h with ghg and g in (5.2.7), as well as Xg with Yghg−1 and Yh with Xg, gives −1 pgh(Yghg−1 ghg ◦φ Xgg) = pgh(Yghg−1 ◦φ Xg)gh.

g−1 Now G-invariance of Y implies immediately Yghg−1 = Yh . So we have

−1 g−1 pgh(Yghg−1 ghg ◦φ Xgg) = pgh(Yh ◦φ Xg)gh, g−1 whence we need to show pgh(Yh ◦φ Xg)gh = pgh(Yh ◦φ Xg)gh. Recall our expressions for Yh and Xg from above, in terms of the vi, uj, Y¯ and X¯. ¯ ¯ In the notation of Lemma 4.3.3, we may assume that ωg|X, and hence that ωg−1 |X by ∗ Lemma 4.1.2. We write Y¯ as a sum of monomials Y1 ··· Ye for functions Yi ∈ V . For g−1 g−1 g−1 ¯ ¯ each Yi and uj we have Yi (uj) = Yi( uj) = Yi(uj). We also have Yi X = YiX ¯ since ωg−1 |X. From these observations and the expression of Lemma 4.2.1, since Yh is a sum of elements of the form v1 ··· vtY , we deduce an equality g−1 X g ¯ g−1 X g ¯ Yh ◦φ Xg = (v1 . . . vt)(Y ) ◦φ Xg = (v1 . . . vt)Y ◦φ Xg. Finally, by the same argument given for the equality (5.2.6) we ﬁnd also that

X g X pgh (v1 . . . vt)Y¯ ◦φ Xg = pgh (v1 . . . vt)Y¯ ◦φ Xg = pgh(Yh ◦φ Xg). Taking these two sequences of equalities together gives the desired equality

g−1 pgh(Yh ◦φ Xg) = pgh(Yh ◦φ Xg), establishes (5.2.5), and completes the proof.

5.3. Corollaries: The codimension grading and some general vanishing results. We apply the formula in Theorem 5.2.3 to analyze distinct cases. In one case, we consider brackets with an element X supported on group elements that act trivially on V . In another case, we consider brackets of X and Y supported on elements that act nontrivially. We will have in this second case some general vanishing results. The following observation helps in organizing these cases. Observation 5.3.1. The graded G-module M g V•−codimV g g ∗ ⊥ S(V ) ⊗ (V ) detg (5.3.2) g∈G LIE BRACKET ON HOCHSCHILD COHOMOLOGY 19 decomposes as a direct sum of graded G-subspaces D(i), for 0 ≤ i ≤ dim(V ), as does its G-invariant subspace,  G M g V•−codimV g g ∗ ⊥ M G  S(V ) ⊗ (V ) detg  = D(i) . (5.3.3) g∈G 0≤i≤dim(V ) The subspace D(i) consists of all sums of elements supported on group elements g for which codimV g = i. We call the decomposition (5.3.3) the codimension grading for HH•(S(V )#G). Said another way, D(i) consists of all summands in (5.3.2) whose first nonzero cohomology class occurs in degree i. Note that classes in D(0) are supported on only those group elements which act trivially on V . The brackets between elements in D(0)G will just be given by the Schouten brackets (cf. [17, Corollary 7.4]): P P Corollary 5.3.4. Let X = g Xgg and Y = h Yhh be G-invariant cocycles in g V•−codimV g g ∗ ⊥ ⊕g∈GS(V ) ⊗ (V ) detg that are supported on group elements acting trivially, i.e. X,Y ∈ D(0). Then in cohomology, X [X,Y ] = {Xg,Yh}gh. g,h g h Proof. In this case for each g, h with Xg and Yh nonzero we will have V = V = gh V = V and pg = id, ph = id and pgh = id. The result now follows from Theo- rem 5.2.3. We refer directly to Theorem 5.2.3 for information on the bracket between cochains in D(0)G and D(> 0)G. We next give some conditions under which brackets are 0. The following corollary was first proved in [17] using different techniques. Corollary 5.3.5 ([17, Proposition 8.4]). Let g, h ∈ G be such that (V g)⊥ ∩ (V h)⊥ is nonzero and is a kG-submodule of V . Let X,Y be G-invariant cocycles in the g V•−codimV g g ∗ ⊥ sum ⊕g∈GS(V ) ⊗ (V ) detg supported on the conjugacy classes of g, h, respectively. Then [X,Y ] = 0. Proof. The hypotheses imply that (V aga−1 )⊥ ∩ (V bhb−1 )⊥ is nonzero for all a, b ∈ G. We will argue that Xgg ◦φ Yhh = 0 at the chain level, and similar reasoning will −1 −1 −1 −1 apply to Xaga−1 aga ◦φ Ybhb−1 bhb and to Ybhb−1 bhb ◦φ Xaga−1 aga . Consider the argument o(yj1 , . . . , yjl−1 , y1, . . . , ym, yjl+1 , . . . , yjd ) in the proof of Lemma 4.2.1. g ⊥ h ⊥ This can be nonzero only in case yjl ∈ (V ) ∩ (V ) , due to linear dependence of the vectors involved otherwise. Thus the only possible terms in the φ-circle product formula of Lemma 4.2.1 that could be nonzero are indexed by such l. However then h Xl(vσ(r)) = 0 for all r, σ, since vσ(r) ∈ V , in the notation of the proof of Lemma 4.2.1. The following corollary was pointed out to us by Travis Schedler. Corollary 5.3.6. For classes X ∈ D(i)G and Y ∈ D(j)G we have [X,Y ] ∈ D(i+j)G. 20 CRIS NEGRON AND SARAH WITHERSPOON

0 0 Proof. As was argued in the proof of Corollary 5.3.5 we ﬁnd that X ◦φY = 0 whenever 0 g V•−codimV g g ∗ ⊥ 0 h V•−codimV h h ∗ ⊥ X ∈ S(V ) ⊗ (V ) detg and Y ∈ S(V ) ⊗ (V ) deth g ⊥ h ⊥ 0 0 and (V ) ∩ (V ) 6= 0. That is to say, nonvanishing of the element X ◦φ Y implies (V g)⊥∩(V h)⊥ = 0 and therefore codimV gh = codimV g +codimV h [16, Lemma 2.1]. It P P follows that for (even non-invariant) classes X = g Xgg ∈ D(i) and Y = h Yhh ∈ D(j) we have X [X,Y ]φ = [Xgg, Yhh]φ ∈ D(i + j). g,h Corollary 5.3.7. The Hochschild cohomology HH•(S(V )#G) is a graded Gersten- haber algebra with respect to the codimension grading. Proof. We just saw in Corollary 5.3.6 that the Gerstenhaber bracket respects the codimension grading. It has also already been established that the cup product respects the codimension grading [3], [16, Proposition 8.1, Theorem 9.2]. The following corollary generalizes [17, Theorem 9.2], where it was proven in homological degree 2. The cocycles X,Y in the corollary are by hypothesis of smallest possible homological degree in their group components. P P Corollary 5.3.8. Take any G-invariant cocycles X = g∈G Xgg, Y = h∈G Yhh in g ⊥ ⊕g∈GS(V ) ⊗ detg that are supported on elements that act nontrivially on V . Then [X,Y ] = 0 on cohomology. Proof. We have that X and Y are sums of invariant classes X0 and Y 0 which are of minimal degrees in D(i) and D(j) respectively, i.e. degrees i and j. By Corollary 5.3.6 we have [X0,Y 0] ∈ D(i+j). But now since the Gerstenhaber bracket is homogeneous of degree −1 we have that [X0,Y 0] is of degree i+j−1. Since there are no nonzero classes 0 0 of degree i + j − 1 in D(i + j) we have that all [X ,Y ] = 0 and hence [X,Y ] = 0.

6. (Non)vanishing of brackets in the case (V g)⊥ ∩ (V h)⊥ = 0 With Corollaries 5.3.5 and 5.3.8 we seem to be approaching a general result. Namely, that for any X and Y supported on group elements that act nontrivially we will have [X,Y ] = 0. It is even known that such a vanishing result holds in degree 2 by [17, Theorem 9.2]. This is, however, not going to be the case in higher degrees. The result even fails to hold when we consider the bracket [X,Y ] of elements in degrees 2 and 3 (see Example 6.1.2 below). We give in this section a few examples to illustrate this nonvanishing, and (re)establish vanishing in degree 2.

6.1. Some examples for which (V g)⊥ ∩ (V h)⊥ = 0. The following two examples illustrate, ﬁrst, the essential role of taking invariants in establishing the degree 2 vanishing result of [17, Theorem 9.2] and, second, an obstruction to establishing a general vanishing result in the case (V g)⊥ ∩ (V h)⊥ = 0. LIE BRACKET ON HOCHSCHILD COHOMOLOGY 21

Example 6.1.1. Take G = Z/2Z × Z/2Z and V = k{x1, x2, x3}. Let g and h be the generators of the ﬁrst and second copies of Z/2Z, and take the G-action on V deﬁned by δi1 δj3 g · xi = (−1) xi and h · xj = (−1) xj. g h So V = k{x2, x3}, (1 − g)V = kx1, V = k{x1, x2}, (1 − h)V = kx3. Obviously, (1 − g)V ∩ (1 − h)V = 0. We also have respective generators

ωg = ∂1, ωh = ∂3, and ωgh = ∂1∂3 of the highest wedge powers of (1−g)V ∗, (1−h)V ∗, and (1−gh)V ∗ respectively. ⊥ ⊥ ⊥ Thus detg = kωgg, deth = kωhh, and detgh = kωghgh. Consider the degree 2 cochains g g ∗ ⊥ X = ωg∂2g ∈ S(V ) ⊗ (V ) detg and h h ∗ ⊥ Y = x2ωh∂2h ∈ S(V ) ⊗ (V ) deth . Then one can easily see, directly from Lemma 4.2.1, that applying the bilinear operation [ , ]φ produces gh gh ∗ ⊥ [X,Y ]φ = ωgωh∂2gh = ωgh∂2gh ∈ S(V ) ⊗ (V ) detgh. This would appear to contradict the degree 2 vanishing result of [17], but it actually R R does not! The point is that neither X nor Y is invariant. In fact, X · G = Y · G = 0. Example 6.1.2. Let G = Z/NZ × Z/MZ for integers N, M > 1. We assume k = k¯, or that M = N = 2. Let σ and τ be the generators of Z/NZ and Z/MZ respectively. Take W = k{x1, x2, x3, x4, x5} and embed G in GL(W ) by identifying σ and τ with the diagonal matrices σ = diag{ζ, ζ−1, 1, 1, 1}, τ = diag{1, 1, 1, ϑ−1, ϑ}, where ζ and ϑ are primitive Nth and Mth roots of 1 in k. We have (1 − σ) = diag{(1 − ζ), (1 − ζ−1), 0, 0, 0}, (1 − τ) = diag{0, 0, 0, (1 − ϑ−1), (1 − ϑ)}, and these are both rank 2 matrices. More speciﬁcally, (1 − σ)W = k{x1, x2} and σ ⊥ τ ⊥ (1 − τ)W = k{x4, x5}. So (W ) ∩ (W ) = 0. Similarly we have στ = diag{ζ, ζ−1, 1, ϑ−1, ϑ}, (1 − στ) = diag{(1 − ζ), (1 − ζ−1), 0, (1 − ϑ−1), (1 − ϑ)},

(1 − στ)W = k{x1, x2, x4, x5}. We take ωσ = ∂1∂2, ωτ = ∂4∂5, ωστ = ∂1∂2∂4∂5, and X = ωσ∂3σ, Y = x3ωτ τ. In this case X and Y are G-invariant, and hence repre- sent classes in HH•(S(V )#G). One then produces via Theorem 5.2.3 the nonvanishing Gerstenhaber bracket 4 [X,Y ] = ωστ στ ∈ HH (S(V )#G). This example can be generalized easily to produce nonzero brackets in higher degree. 22 CRIS NEGRON AND SARAH WITHERSPOON

6.2. Vanishing of brackets in degree 2. Consider the subcomplex

M g V•−codimV g g ∗ ⊥ D(1) ⊂ S(V ) ⊗ (V ) detg g consisting of all summands corresponding to group elements g with codimV g = 1. This subcomplex is stable under the G-action. It was seen already in [4, 17] that D(1)G = 0. So actually, after we take invariance, we have • M g V•−codimV g g ∗ ⊥ G G G HH (S(V )#G) = S(V ) ⊗ (V ) detg = D(0) ⊕ D(> 1) . g where in D(> 1) we have all summands corresponding to g with codimV g > 1. It follows that, after we take invariants and restrict ourselves to considering only elements in degree 2, the only situations that can occur when taking brackets in D(> 0)G = D(> 1)G are covered by Corollary 5.3.8. Hence when we apply the Gerstenhaber bracket we get

h 2G 2Gi D(> 0) , D(> 0) = 0.

This rephrases the argument given in [17, Theorem 9.2].

References [1] R. O. Buchweitz, E. L. Green, N. Snashall, and Ø. Solberg. Multiplicative structures for Koszul algebras. Q. J. Math., 59(4):441–454, 2008. [2] V. Dolgushev and P. Etingof. Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology. Int. Math. Res. Not., 2005(27):1657–1688, 2005. [3] P. Etingof and V. Ginzburg. Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math., 147(2):243–348, 2002. [4] M. Farinati. Hochschild duality, localization, and smash products. J. Algebra, 284(1):415–434, 2005. [5] M. Gerstenhaber and S. D. Schack. Relative Hochschild cohomology, rigid algebras, and the Bockstein. J. Pure Appl. Algebra, 43(1):53–74, 1986. [6] G. Ginot and G. Halbout. A formality theorem for Poisson manifolds. Lett. Math. Phys., 66(1- 2):37–64, 2003. [7] V. Ginzburg and D. Kaledin. Poisson deformations of symplectic quotient singularities. Adv. Math., 186(1):1–57, 2004. [8] G. Halbout and X. Tang. Noncommutative Poisson structures on orbifolds. Trans. Amer. Math. Soc., 362(5):2249–2277, 2010. [9] G. Hochschild, B. Kostant, and A. Rosenberg. Differential forms on regular affine algebras. Trans. Amer. Math. Soc., 102:383–408, 1962. [10] M. Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math. Phys., 66(3):157–216, 2003. [11] W. Lowen and M. Van den Bergh. Hochschild cohomology of abelian categories and ringed spaces. Adv. Math., 198(1):172–221, 2005. [12] C. Negron. The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings. J. Pure Appl. Algebra, 221(5):1112–1133, 2017. [13] C. Negron and S. Witherspoon. An alternate approach to the Lie bracket on Hochschild cohomology. Homology Homotopy Appl., 18(1):265–285, 2016. [14] N. Neumaier, M. Pflaum, H. Posthuma, and X. Tang. Homology of formal deformations of proper étaleLie groupoids. J. Reine Angew. Math., 593:117–168, 2006. LIE BRACKET ON HOCHSCHILD COHOMOLOGY 23

[15] M. Pﬂaum, H. Posthuma, X. Tang, and H.-H. Tseng. Orbifold cup products and ring structures on Hochschild cohomologies. Commun. Contemp. Math., 13(1):123–182, 2011. [16] A. V. Shepler and S. Witherspoon. Finite groups acting linearly: Hochschild cohomology and the cup product. Adv. Math., 226(4):2884–2910, 2011. [17] A. V. Shepler and S. Witherspoon. Group actions on algebras and the graded Lie structure of Hochschild cohomology. J. Algebra, 351:350–381, 2012.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA E-mail address: [email protected]

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA E-mail address: [email protected]